Solution Of Direct Problem Research Articles

Direct and inverse analysis of diffusive logistic population evolution with time delay and impulsive culling via integral transforms and hybrid optimization

Motivated by the fact that several species act as a vector in the spread of human or livestock diseases, many works propose mathematical formulations for the modeling of these populations, most of them considering Fickian dispersion and logistic like growth rates. For the best use of these models in a real application of optimal population control, the model parameters should be identified as accurately as possible for a given species population. In this work, this parameter identification problem is formulated as an inverse problem, which is tackled with a combination of the Generalized Integral Transform Technique (GITT), for the direct problem solution, and a hybrid stochastic–deterministic procedure for the minimization of the defined objective function in the inverse analysis, employing the Differential Evolution and the Levenberg–Marquardt methods. The direct problem solution with GITT and the inverse analysis are critically investigated. In order to improve the computational performance of the inverse problem solution, a second order semi-analytical integration and a solution refinement scheme are proposed.

Determining the focal mechanisms of the events in the Carpathian region of Ukraine

Abstract. The modification of the matrix method for constructing the displacement field on the free surface of an anisotropic layered medium is presented. The source of seismic waves is modelled by a randomly oriented force and seismic tensor. A trial and error method is presented for solving the inverse problem of determining parameters of the earthquake source. A number of analytical and numerical approaches to determining the earthquake source parameters, based on the direct problem solutions, are proposed. The focal mechanisms for the events in the Carpathian region of Ukraine are determined by the graphical method. The theory of determination of the angles of orientation of the fault plane and the earthquake's focal mechanism are presented. The focal mechanisms obtained by two different methods are compared.

L(α)-stable variable-order implicit second derivative Runge-Kutta methods

This paper considers the extension of the popular Runge-Kutta methods (RKM) to second derivative Runge-Kutta methods (SDRKMs) for the direct solution of stiff initial value problems (IVPs) of ordinary differential equations (ODEs). The methods are based on using collocation and interpolation technique. The last stage of the input approximation is identical to the output method. The SDRKMs are L(α)-stable for the methods examined. Numerical experiments are given comparing one of these methods with a two derivative Runge Kutta method (TDRKM) and a second derivative linear multistep method (SDLMM).

Numerical Solution of Inverse Problems of Thermoelasticity for a Composite Cylinder

The thermoelastic state of a composite cylinder is analyzed. Classical generalized problems defined on classes of discontinuous functions are presented. Explicit expressions are obtained for residual gradients (using the solution of direct and adjoint problems) for the implementation of Alifanov’s gradient methods; functions of the finite element method are used to construct highly accurate computation schemes for the numerical discretization of direct and adjoint problems.

Solution of Direct and Inverse Problems for Infiltration and Contaminant Adsorption in Partially Saturated Porous Media

We discuss the numerical modelling of unsaturated-saturated flow in porous media and contaminant transport with adsorption in 1D. We developed an efficient numerical approximation which is a good candidate for solving inverse problems involving the determination of model parameters (hydraulic and geochemical). Also an adsorption/desorption model for mass transfer through the boundary of porous media (contact solid-liquid) is discussed, too. Our numerical solution is based on the MOL method where space discretization leads to the corresponding system of ODE. To collect the measurement data for inverse problems we consider both the gravitational and centrifugal infiltration driven forces. In series of numerical experiments we demonstrate the effectiveness of our method.

The Direct Kinematics Solution of the 6-RSPS Simulator Platform with BP Networks

A new structure of 6-RSPS simulator platform which can be rotated infinitely is presented. The solution of direct kinematics problem of parallel structure is the fundamental problem. For the solution of this problem, according to the links vector diagram, the relationship of the position and orientation of the platform between the length of links and the rotate angle of the under-hinge is derived. BP network is applied, then using the previous data obtained by inverse kinematics as the training data and testing data of BP network, the direct kinematics solution is obtained. Simulation results verified that the solution of direct kinematics applied the BP network can be meet the requirements completely and the error percentages are within the acceptable range.

Temperature distributions at the surface of functionally graded materials containing a cylindrical defect

The recognition of subsurface defects is an important aspect of inverse heat transfer problems. The solution of direct problems is the basis of the inverse problems. In this paper, based on the non-Fourier heat conduction equation and using the image method and the wave function expansion method, an analytical method to study the multiple scattering of thermal waves and temperatures from the subsurface defect in semi-infinite functionally graded materials (FGMs) is presented. A general solution of the scattered fields of thermal waves is obtained. Without loss of generality, the incidence of thermal waves excited by the periodically modulated laser and the subsurface defect are treated as the line heat source and a cylindrical cavity with thermal insulation in the modelling, respectively. The amplitude of the changes in temperature at the frontal surface in the cylindrical defect due to different physical and geometrical parameters is analysed, discussed, and presented graphically using the analytical method. The significant changes in the temperature were found to occur at the frontal surface because of the existence of conducting defects. In addition, the influence of the defect state on the temperature was found to also be related to the nonhomogeneous parameters of FGMs, the wavenumber, the frequency of the thermal waves, and the depth of the buried defects. The numerical results and method proposed in this paper are expected to be applied for the analysis of infrared thermal waves in non-destructive evaluations of functionally graded materials and inverse problems of mathematical physics.

Inverse internal pressure estimation of functionally graded cylindrical shells under thermal environment

The time-dependent pressure in long functionally graded cylindrical shells under thermal environment is estimated using the measured strains on their outer surfaces in conjunction with an inverse algorithm. The obtained strains from the solution of the related direct problem are used to simulate the data measurements. The direct problem is formulated based on the three-dimensional thermoelasticity theory under the plane strain conditions. The inverse solution procedure benefits from the discrepancy principle together with the conjugate gradient method as a powerful technique for optimization procedure. The differential quadrature method as an efficient and accurate numerical tool is employed to discretize the governing differential equations subjected to the related boundary and initial conditions in both spatial and temporal domains. The influence of measurement errors on the accuracy of the estimated internal pressure and also the displacement and stress components is investigated. The good accuracy of the results validates the presented approach.

Solution of direct kinematic problem for Stewart-Gough platform with the use of analytical equation of plane

The paper concerns the solution of direct kinematic problem for the Stewart-Gough platform of the type 6-3. The article represents a detailed analysis of methods of direct kinematic problem solution for platform mechanisms based on the parallel structures. The complexity of the problem solution is estimated for the mechanisms of parallel kinematics in comparison with the classic manipulators, characterized by the open kinematic chain. The method for the solution of this problem is suggested. It consists in setting up the correspondence between the functional dependence of Cartesian coordinates and the orientation of the moving platform centre on the values of generalized coordinates of the manipulator, which may be represented, in the case of platform manipulators, by the lengths of extensible arms to connect the foundation and the moving platform of the manipulator. The method is constructed in such a way that the solution of the direct kinematic problem reduces to solution of the analytical equation of plane where the moving platform is situated. The equation of the required plane is built according to three points which in this case are attachment points of moving platform joints. To define joints coordinates values it is necessary to generate a system of nine nonlinear equations. It ought to be noted that in generating a system of equation are used the equations with the same type of nonlinearity. The physical meaning of all nine equations of the system is Euclidean distance between the points of the manipulator. The location and orientation of the moving platform is represented as a homogeneous transformation matrix. The components of translation and rotation of this matrix can be defined through the required plane. The obtained theoretical results are supposed to be used in the decision support system during the complex research of multi-sectional manipulators of parallel kinematics to describe the geometrically similar 3D-prototype of the manipulator. In the process of work it is planned to make the optimal choice of numerical approach to the solution of the suggested equation system, considering the nonlinearity type etc., to develop the methodology for evaluation of received solutions, to adjust the suggested method for the solution of direct kinematic problem for the Stewart-Gough platform of the type 6-6.

Numerical Solution of Direct and Inverse Scattering Problems in 1D with a General Nonlinearity

Abstract. We consider direct and inverse scattering problems for a general nonlinear Schrödinger equation. The direct problem is solved by the method of integral equations. The inverse problem is solved by the method of Born approximation. The computation of the Born approximation is carried out using regularization methods. Numerical examples are given to illustrate the effectiveness of these methods.

On a New Formulas for a Direct and Inverse Cauchy Problems of Heat Equation

In this paper a solution of the direct Cauchy problems for heat equation is founded in the form of Hermite polynomial series. A well-known classical solution of direct Cauchy problem is represented as Poisson's integral. The author reveals, the formulas obtained by him for solution of the inverse Cauchy problems have a symmetry with respect to the formulas for corresponding direct Cauchy problems. Obtained formulas for solution of the inverse problems can serve as a basis for reg-ularizing computational algorithms while well-known classical formula for the solution of the inverse Cauchy problem can't be a basis for regu-larizing computational algorithms.

Direct Solution of Initial Value Problems of Fourth Order Ordinary Differential Equations Using Modified Implicit Hybrid Block Method

Our focus in this article is the derivation; analysis and implementation of a new modified implicit hybrid block method for the direct solution of initial value problems of fourth order ordinary differential equations. In the derivation of the method, we adopted the approach of collocation approximation to obtain the main scheme with continuous coefficients. From the main scheme, additional schemes were developed. The implementation strategy of the new method is by combining the main scheme and the additional schemes as

Approaches to accommodate noisy data in the direct solution of inverse problems in incompressible plane strain elasticity

We apply the adjoint-weighted equation method (AWE) to the direct solution of inverse problems of incompressible plane strain elasticity. We show that based on untreated noisy displacements, the reconstruction of the shear modulus can be very poor. We link this poor performance to loss of coercivity of the weak form when treating problems with discontinuous coefficients. We demonstrate that by smoothing the displacements and appending a regularization term to the AWE formulation, a dramatic improvement in the reconstruction can be achieved. With these improvements, the advantages of the AWE method as a direct solution approach can be extended to a wider range of problems.

Reconstruction of the Discontinuity Line of a Piecewise-Constant Coefficient in the Two-Dimensional Internal Initial–Boundary Value Problem for the Homogeneous Heat Equation

We investigate the reconstruction of the discontinuity line of a piecewise-constant coefficient in the two-dimension internal initial---boundary value problem for the one-dimensional heat equation. Supplementary data include the direct problem solution, which is known for finitely many boundary points at all times. The results of computer experiments reported in the article show that the inverse problem is well-conditioned in this setting. The direct problem has been reduced to the boundary-value problem for the Helmholtz equation and its solution was expressed in terms of potentials.

The solution of direct and inverse transient heat conduction problems with layered materials using exponential basis functions

In this paper we present an extension of the formulation given in (EABE, 37 (2013) 868–883) to direct and inverse heat conduction problems in layered materials. Expressing the numerical solution as a series of exponential basis functions (EBFs) defined in space and time, is the main idea of the presented method. We shall show that the use of the EBFs provides a versatile tool for the solution of a variety of problems such as the classical heat diffusion with internal heat sources and the inverse problems in layered media with perfect and imperfect contacts. To this end, we choose a proper set of EBFs satisfying the governing differential equation and interface conditions. Then a collocation technique is employed to satisfy both initial and boundary conditions. The proposed formulation can also be easily extended to the solution of non-Fourier heat conduction problems or the analysis of thermal wave propagation. The capability of the presented method is investigated in the solution of some direct and inverse heat conduction problems including a boundary identification problem and backward heat conduction problems. Issues pertaining to short time solution and penetration time, in inverse heat conduction problems, are addressed in the problems solved.

Solution of forward kinematics in Stewart platform using six rotary sensors on joints of three legs

A novel method is proposed for real-time solution of direct kinematics problem of Stewart platform (SP) using six measurements on three legs’ joints consisting of the rotations of three legs in two directions. After the application of the method on a laboratory sample SP, it is observed that the method is preferable to the conventional method that uses the length measurements of all six legs, in the grounds of industrial applicability. It is due to simpler implementation, less expense, easier maintenance, and stress-free assembly. The algorithms of both forward and inverse kinematics are fully derived based on geometric relationships between the platform states and the measurement data. The sensitivity to the measurement errors is analyzed theoretically and is applied through a computer simulation to several configurations of the sample SP which are uniformly distributed in the workspace. The variances of measurement errors for those configurations are compared between the conventional and proposed methods and it is observed that: the proposed method operates more accurate in position measurement especially in lateral movements. Additionally, the proposed method is not too sensitive to direction of movement and geometry of the SP.

Aerospace laser sensing of cloudiness: numerical statistical modeling

In the numerical modeling of laser radiation transfer in optically dense cloudy media it is necessary to take into account multiple scattering effects, which alter the spatiotemporal structure of light pulses. The Monte Carlo method makes it possible to achieve the most complete account of these effects in the solution of direct problems of laser sensing of scattering media. This work considers two problems. The first is connected with construction of an adequate optical model of crystalline clouds which takes account their optical anisotropy. The second touches on questions of Monte Carlo modeling of laser radiation transfer in optically anisotropic media. A number of results of numerical experiments are presented which establish a quantitative connection between some cloud parameters and the magnitude and shape of the time convolution of a non-stationary laser return signal reflected by a single-layer continuous crystalline or liquid-droplet cloud and by two-level continuous cloudiness, when the crystalline cloud is located above the liquid-droplet cloud.

Rock destruction by tension

The article describes a procedure proposed for solving inverse problems on identification of flat specimen fracture by tension using in situ displacements and acoustic emission. Weak associated area with concentration of pores is modeled by a mathematical cut with lengthwise averaged normal tensile stresses. In places with high concentration of pores, Young’s modulus is “degraded,” i.e. strains arise here and cause shear stresses. The latter are calculated by successive approximations of solutions of direct problems by specimen contour displacements and data on acoustic emission. In principle, under consideration are inverse problems of the continuum mechanics.

New approach for modelling of transiting exoplanets for arbitrary limb-darkening law

We present a new solution of the direct problem of planet transits based on transformation of double integrals to single ones. On the basis of our direct problem solution, we created the code tac-maker for rapid and interactive calculation of synthetic planet transits by numerical computations of the integrals. The validation of our approach was made by comparison with the results of the wide-spread Mandel–Agol method for the cases of linear, quadratic and squared-root limb-darkening laws and various combinations of model parameters. For the first time our approach allows the use of arbitrary limb-darkening law of the host star. This advantage together with the practically arbitrary precision of the calculations makes the code a valuable tool that faces the challenges of the continuously increasing photometric precision of the ground-based and space observations.

Piecewise Polynomial Control Parameterization in the Direct Solution of Optimal Control Problems

This paper describes an algorithm for the direct solution of a class of optimal control problems. The algorithm is based on approximating the unknown control inputs via a finite dimensional parameterization. Specific control approximations that are implemented include (i) piecewise constant, (ii) piecewise linear, or (iii) piecewise cubic polynomials. The cubic approximation presented here is believed to be new. Another novel feature of the algorithm is that the state equations are approximated using single step Runge–Kutta methods on a fixed mesh. The parameterized optimal control problem is solved using a sequential quadratic programming technique. The paper presents examples to illustrate the convergence behavior of the various control parameterization schemes.